{"title":"Concentration phenomena of solutions for critical perturbed Hénon problems","authors":"","doi":"10.1016/j.jde.2024.08.046","DOIUrl":null,"url":null,"abstract":"<div><p>The main aim in this paper is to carry out a comprehensive research on the critical Hénon problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow></msup><mo>+</mo><mi>ϵ</mi><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mspace></mspace><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mspace></mspace><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace><mspace></mspace><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mi>α</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>, <span><math><mi>q</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span>, Ω is a smooth bounded domain containing the origin in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>. Based on Lyapunov-Schmidt reduction argument, we provide some sufficient conditions for the existence of concentrating solutions without any condition on the Robin function. The main results depend on the non-resonant case that <span><math><mi>k</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>≠</mo><mn>0</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>q</mi><mo>≠</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> and the resonant case that <span><math><mi>k</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext>or</mtext><mspace></mspace><mi>q</mi><mo>=</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>. The novelty in our study is significantly different from the case that <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span>.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002203962400528X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The main aim in this paper is to carry out a comprehensive research on the critical Hénon problem where , , , , , Ω is a smooth bounded domain containing the origin in , . Based on Lyapunov-Schmidt reduction argument, we provide some sufficient conditions for the existence of concentrating solutions without any condition on the Robin function. The main results depend on the non-resonant case that and the resonant case that . The novelty in our study is significantly different from the case that .
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics